Prime Submodules and Flat Modules

In this paper, some characterizations of prime submodules in flat modules and, particularly, in free modules are given. Furthermore, the height of prime submodules and some saturated chain of prime submodules are also given.     

Strongly Irreducible Submodules of Modules

Strongly irreducible submodules of modules are defined as follows： A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N or K ∈ N. The relationship among the families of irreducible, strongly irreducible, prime and primary submodules of an R-module M is considered, and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given.     

Modules with Chain Conditions on Non-essential Submodules

Abstract

We investigate when modules which satisfy the descending (respectively, ascending) chain condition on non-essential submodules are uniform or Artinian (respectively, Noetherian).     

Supplement Submodules of Injective Modules

An R-module M is called almost injective if M is a supplement submodule of every module which contains M. The module M is called F-almost injective if every factor module of M is almost injective. It is shown that a ring R is a right H-ring if and only if R is right perfect and every almost injective module is injective. We prove that a ring R is semisimple if and only if the R-module R _R is F-almost injective.     

On Pure Quotients and Pure Subobjects

In the theory of accessible categories, pure subobjects, i.e. filtered colimits of split monomorphisms, play an important role. Here we investigate pure quotients, i.e., filtered colimits of split epimorphisms. For example, in abelian, finitely accessible categories, these are precisely the cokernels of pure subobjects, and pure subobjects are precisely the kernels of pure quotients.     

可补闭子模和有界广义逆模映射

引入了“有界广义逆模映射”的概念 ,并给出了等价刻画 .揭示了有界广义逆模映射和可补闭子模的内在联系 .     

Closed Submodules of Holomorphic Functions with Two Generators

Let I be a closed submodule over a polynomial ring in a space of holomorphic functions on a domain in the complex plane. We establish sufficient conditions under which I is generated by two functions or two special submodules. As a corollary, it follows from these results that if an invariant subspace W C (a,b) (with respect to the differentiation operator) admits spectral synthesis, then it is the solution space of a system of two homogeneous convolution equations.     

Basic Submodules of Modules over Serial, Right Noetherian Rings. I

It is proved that any right module over a serial, right Noetherian ring R contains a basic submodule. The structure of submodules of an indecomposable pure injective R-module is investigated. Bibliography: 11 titles.     

Basic Submodules of Modules Over Serial, Right Noetherian Rings. II

In the preceding paper, the authors published an existence theorem for basic submodules of right modules over right Noetherian, serial rings. The aim of the present paper is to prove a uniqueness theorem for basic submodules over such rings. Bibliography: 13 titles.     

The Admissibility of Simple Bounded Modules for an Affine Lie Algebra

This paper studies a class of simple integrable modules for an affine Lie algebras which are closely related to the finite-dimensional modules studied by V. Chari and A. Pressley, except that the Euler element is assumed to act. They are infinite-dimensional; but are shown to have finite-dimensional weight spaces. It is conjectured that any simple integrable module with a zero weight space belongs to this class and their classification is given. The main interest in studying such modules is that they may occur in the endomorphism rings of highest weight modules whilst those of Chari and Pressley in general do not. Their character theory is also more complicated.     

Characters of Simple Bounded Modules over an Untwisted Affine Lie Algebra

After V. Chari and A. Pressley, a simple integrable module with finite-dimensional weight spaces over an affine Lie algebra is either a standard module (highest or lowest weight), in which case its formal character is given by the famous Weyl–Kac formula, or a subquotient of a tensor product of loop modules. In this paper we compute formal characters of generic simple integrable modules of the latter type.     

Classification of Hardy Submodules and Characteristic Space Theory（A Brief and Selective Survey）

This paper is a brief and selective survey on classification of Hardy Submodules over the polydisk. The survey reports some progress in classification of Hardy submodules.     

The Factor Decomposition Theorem of Bounded Generalized Inverse Modules and Their Topological Continuity

In this paper we obtain a Douglas type factor decomposition theorem about certain important bounded module maps. Thus, we come to the discussion of the topological continuity of bounded generalized inverse module maps. Let X be a topological space, x →Tx ： X→L（E） be a continuous map, and each R（Tx） be a closed submodule in E, for every fixed x C X. Then the map x→ Tx^＋： X→L（E） is continuous if and only if ｜｜Tx^＋｜｜ is locally bounded, where Tx^＋ is the bounded generalized inverse module map of Tx. Furthermore, this is equivalent to the following statement： For each x0 in X, there exists a neighborhood ∪0 at x0 and a positive number λ such that （0, λ^2）lohtatn in ∩x∈∪0C／σ（Tx^＋Tx）, where a（T） denotes the spectrum of operator T.     

Characterization of Certain Holomorphic Geodesic Cycles on Quotients of Bounded Symmetric Domains in terms of Tangent Subspaces

Let be an irreducible bounded symmetric domain and Aut() be a torsion-free discrete group of automorphisms, X /. We study the problem of algebro-geometric and differential-geometric characterizations of certain compact holomorphic geodesic cycles S X. We treat special cases of the problem, pertaining to a situation in which S is a compact holomorphic curve, and to the case where is a classical domain dual to the hyperquadric. In both cases we consider algebro-geometric characterizations in terms of tangent subspaces. As a consequence we derive effective pinching theorems where certain complex submanifolds S X are proven to be totally geodesic whenever their scalar curvatures are pinched between certain computed universal constants, independent of the volume of the submanifold S, giving new examples of the gap phenomenon for the characterization of compact holomorphic geodesic cycles.Research funded by a CERG grant from Research Grants Council of Hong Kong.     

Anchor Maps and Stable Modules in Depth Two

An algebra extension A ∣ B is right depth two if its tensor-square is in the Dress category . We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of two-sided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu, left and right bialgebroids over a base algebra R may be defined in terms of anchor maps, or representations on R. The anchor maps for the bialgebroids and over the centralizer R = C _A (B) are the modules _S R and R _T studied in Kadison (J. Alg. & Appl., 2005, preprint), Kadison (Contemp. Math., 391: 149–156, 2005), and Kadison and Külshammer (Commun. Algebra, 34: 3103–3122, 2006), which provide information about the bialgebroids and the extension (Kadison, Bull. Belg. Math. Soc. Simon Stevin, 12: 275–293, 2005). The anchor maps for the Hopf algebroids in Khalkhali and Rangipour (Lett. Math. Phys., 70: 259–272, 2004) and Kadison (2005, preprint) reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in Van Oystaeyen and Panaite (Appl. Categ. Struct., 2006, in press). We sketch a theory of stable A-modules and their endomorphism rings and generalize the smash product decomposition in Kadison (Proc. Am. Math. Soc., 131: 2993–3002, 2003 Prop. 1.1) to any A-module. We observe that Schneider’s coGalois theory in Schneider (Isr. J. Math., 72: 167–195, 1990) provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra. A homomorphism of finite dimensional coalgebras is codepth two if and only if its dual homomorphism of algebras is depth two.